Am 12.12.2010 06:35, schrieb Darren New:

> Yeah, except you don't start out with "things". That's why it's like
> that 1-function variant Andrew always talks about. The S or the K or
> whatever. The only "thing" you have is a set. And sets don't have order.
> So how do you even *represent* "three" using only sets, with nothing in
> the sets but other sets?

Oh, that's pretty simple:

First, we start with a contrived unary operation on arbitrary sets we 
call "increment" some obscure reason:

   A++ := A U {A}

(The notation for the operation is, as you may guess, not common in the 
mathematical world ;-))

Next, we define the inverse operation:

   (A++)-- := A

Next, we define a set we call "zero"; we could use an arbitrary 
definition, but as we have no other sets in our little world yet, we use 
the empty set:

   0 := {}

Next, we recursively define a set of objects we call "natural numbers":

   N  := { n | n-- is element of N U 0}
   N° := N U 0

Now we can assign symbols to the natural numbers:

   3  := { {}, {{}}, {{}, {{}}} }

There.

 From here on, we can extend our little world of arithmetics by defining 
an operation on two arbitrary natural numbers called "addition", define 
the inverse operation called "subtraction", define another operation 
called "multiplication", and so forth...

And it all lives within the realms of set theory.

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